19.1.Linearity and Causality
We have frequently described media as linear, meaning by this that D(t), for example, was related to E(t) by a constant of proportionality called the permittivity. In dispersive media, such a relationship is not possible, but the concept of a linear medium is still very useful.
19.2.Frequency Response and Dispersion
The first application of the dispersion relations was the determination of the index of refraction of a solid material for X-rays. In this case, it is relatively easy to measure the absorption of the radiation by the material, but very difficult to measure its refraction. The index of refraction can, however, be determined by using the dispersion relations and the measured X-ray absorption. We now turn to an examination of dispersive media, where we shall find occasion to illustrate the use of the dispersion relations.
19.3.Drude-Lorentz Harmonic Oscillator Model
All ordinary matter is composed of negative electrons and positive nuclei. If, for the purpose at hand, some of the electrons (more or fewer than Z, the nuclear charge) can be considered as tightly bound to the nucleus and moving with it, the composite entity is a charged ion. The electrons or ions will be treated as harmonic oscillators—that is, particles bound to an equilibrium position by a linear restoring force. For generality, we make it a damped harmonic oscillator, including a linear damping force proportional to the velocity. When an electromagnetic wave is present, the oscillator is driven by the electric field of the wave. The response of the medium is obtained by adding up the motions of the particles; since the assumed forces are linear, the K and g that result from the model will be constant (i.e., independent of E, although they will depend on frequency). Applied to electrons, the model describes the lound electrons in atoms, but free electrons can be included as a special case simply by putting the restoring-force constant of the oscillator equal to zero.
19.4.Resonance Absorption by Bound Charges
The absorption peak is similiar to the electronic one, but it is located in the infrared instead of the visible or the ultraviolet. The corresponding Kr or n makes no contribution at higher frequency, but it does at lower frequency. Thus for rocksalt the static dielectric constant is about 6, in comparison to about 2(= 1.52) in the visible. The letter is due to the electronic absorption in the ultraviolet; the former includes the effect of the ionic absorption in the infrared as well.
19.5.The Drude Free-Electron Theory
All the result of this section also apply when the charged particles are heavy ions instead of electrons. Since the medium has been assumed to be electrically neutral, positive ions must always be present with an average number density N equal to that of the electrons. In metals the positive ions are not freely mobile, of course, but in gaseous plasmas they are, and their motion is often important.
19.6.Dielectric Relaxation
Electrical resonance absorption does not usually occur in materials at frequencies lower than the heavy-ion peaks in the infrared (though it does in manmade structures, of course). There is, however, another kind of absorption mechanism, known as dielectric loss, which can occur at lower frequencies (but not at higher). It is frequently important as a loss mechanism at microwave frequencies and below, and its concomitant dispersion of the real dielectric constant explains, for example, the difference between the static dielectrik constant of water, 81, and the optical value about 1.8(1.332).
19.7.Ocillating Fields in Disversive Media
We have now shown that for mochromatic fields the real permitivity and real conductivity of a linear medium generally depend on the frequency. Furthermore, in the complex representation , this frequency dependance can be described either by a permitivity or by a conductivity that is a complex function of the frequency. For many materials there are, however, large ranges over which the frequency dependence is small enough to be neglected in most application.
In the physical world electromagnetic fields come in wave trains that are, in some cases, sufficienly long that they may be treated as monochromatic to an adequate approximation. Often, however, the wave train is so short that the monochromatic approximation is not adequate. One way of dealing with short pulses is to write the dielctric field, for example, as asuperposition of monochromatic fields. Solutions can be built up this way because Maxwell’s equation are linear as long as we consider only linear materials.
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